Man, I love indoor track! I think you'll have fun with this, although I'd be dreading a 4×2.
That's quite a lineup of races they have you doing. Hopefully, there is a decent amount of time between each of those. My main suggestion is going to be based on this. Assuming you're not waiting hours between each event, you need to adjust your warmup and cooldown for each race accordingly. My suggestion would be to warm up as you normally would before the first race. After the race, get around a mile of easy running in, do some stretching to keep your legs loose, and then get your legs up for a while while you refuel (if you feel the need) on something light and rehydrate. Before the following races, you'll probably need a bit less of easy running for the warmup. After the final race, go ahead and do your regular cooldown.
As for indoor track in general, most tracks are 200 meters long. This does make for some tight turns but they are normally not so tight that, at 6:00 pace give or take, they would be hard to navigate. The turns never bothered me other than occasionally getting a sore left ankle even when running sub-5 pace. It can be difficult to pass on the turns, so you may need to focus on making use of the straights (which, of course, are pretty short themselves) to pass. On the flip side, if you're trying to hold off someone who's passing you, you can usually take advantage of this fact by trying to not let them pass you on the straights and force them to make the pass on the turns where it is harder to do so. The surface is usually some kind of rubber surface, usually not the same kind of surface as an outdoor track but generally with similar properties. Most indoor tracks have good traction and are not overly soft but not as hard as a paved surface.
The starts for the 1500 and 3000 should be a waterfall start. Watch yourself on these, especially if you're on the inside. It's easy to get cut off. You may want to get out a little harder than you normally would in order to avoid finding yourself in a bit of a squeeze. Also, don't be afraid to make contact with people. Depending on how competitive the people in your race are, you may have to use physical contact to defend your space.
As for the 800, it won't be in lanes the whole way. I can't recall how indoor 800s were normally run when I was in school (I never ran one myself) but I do recall they usually started in lanes (I think there may have been an occasional waterfall start but these can get ugly in an indoor 800) and I seem to recall them running in lanes for 1-2 turns before breaking for lane 1. There should be a line on the track and they may have some cones set up where you can break. The instructions before the start of the race should clarify a lot. Starting blocks are not used for the 800, people use a standing start that is essentially like any other distance race.
As far as the 4×200 goes, hopefully you won't be the leadoff leg and have to worry about starting blocks. Even if you are the leadoff leg, if you don't know how to use blocks, my suggestion would be don't use them. Instructions for handing off should be given. You'll have an exchange zone that, as I recall, is 20 meters long. The baton has to be passed within that zone. Without practicing, you'll probably just turn your shoulders and watch the handoff as you take it like 4×400 runners usually do. Doing it this way, it's pretty simple. When taking the baton, you let your teammate hand it to you, then you take it from them and run. When handing off, you set the baton in your teammate's hand and let them take it from you. Then, you get out of the way usually by stepping to the inside of the track so nobody behind you runs you over.